We modify the payment rule of the standard divide the dollar (DD) game by introducing a second stage and thereby resolve the multiplicity problem and implement equal division of the dollar in equilibrium. In the standard DD game, if the sum of players’ demands is less than or equal to a dollar, each player receives what he demanded; if the sum of demands is greater than a dollar, all players receive zero. We modify this second part, which involves a (...) harsh punishment. In the modified game $(D\!D^{\prime })$ , if the demands are incompatible, then players have one more chance. In particular, they play an ultimatum game to avoid the excess. In the two-player version of this game, there is a unique subgame perfect Nash equilibrium in which players demand (and receive) an equal share of the dollar. We also provide an $n$ -player extension of our mechanism. Finally, the mechanism we propose eliminates not only all pure strategy equilibria involving unequal divisions of the dollar, but also all equilibria where players mix over different demands in the first stage. (shrink)

It is sometimes the case that one solution concept in game theory is equivalent to applying another solution concept to a modified version of the game. In such cases, does it make sense to study the former separately, or should we entirely subordinate it to the latter? The answer probably depends on the particular circumstances, and indeed the literature takes different approaches in different cases. In this article, I consider the specific example of Stackelberg mixed strategies. I argue that, even (...) though a Stackelberg mixed strategy can also be seen as a subgame perfect Nash equilibrium of a corresponding extensive-form game, there remains significant value in studying it separately. The analysis of this special case may have implications for other solution concepts. (shrink)

Counterexamples to two results by Stalnaker (Theory and Decision, 1994) are given and a corrected version of one of the two results is proved. Stalnaker's proposed results are: (1) if at the true state of an epistemic model of a perfect information game there is common belief in the rationality of every player and common belief that no player has false beliefs (he calls this joint condition ‘strong rationalizability’), then the true (or actual) strategy profile is path equivalent to (...) a Nash equilibrium; (2) in a normal-form game a strategy profile is strongly rationalizable if and only if it belongs to C8 , the set of profiles that survive the iterative deletion of inferior profiles. (shrink)

We generalize the concept of Nash equilibrium in mixed strategies for strategic form games to allow for ambiguity in the players' expectations. In contrast to other contributions, we model ambiguity by means of so-called lower probability measures or belief functions, which makes it possible to distinguish between a player's assessment of ambiguity and his attitude towards ambiguity. We also generalize the concept of trembling hand perfect equilibrium. Finally, we demonstrate that for certain attitudes towards ambiguity it (...) is possible to explain cooperation in the one-shot Prisoner's Dilemma in a way that is in accordance with some recent experimental findings. (shrink)

We add a stage to Nash’s demand game by allowing the greedier player to revise his demand if the demands are not jointly feasible. If he decides to stick to his initial demand, then the game ends and no one receives anything. If he decides to revise it down to 1-x\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$1-x$$\end{document}, where x is his initial demand, the revised demand is implemented with certainty. The implementation probability changes linearly between these (...) two extreme cases. We derive a condition on the feasible set under which the two-stage game has a unique subgame perfect equilibrium. In this equilibrium, there is first-stage agreement on the egalitarian demands. We also study two n-player versions of the game. In either version, if the underlying bargaining problem is “divide-the-dollar,” then equal division is sustainable in a subgame perfect equilibrium if and only if the number of players is at most four. (shrink)

A new class of values combining marginalistic and egalitarian principles is introduced for cooperative TU-games. It includes some modes of solidarity among the players by taking the collective contribution of some coalitions to the grand coalition into account. Relationships with other class of values such as the Egalitarian Shapley values and the Procedural values are discussed. We propose a strategic implementation of our class of values in subgame perfect Nash equilibrium. Two axiomatic characterizations are provided: one (...) of the whole class of values, and one of each of its extreme points. (shrink)

We study the performance of heuristics relative to the performance of optimal solutions in the rich domain of sequential search, where the decision to stop the search depends only on the applicant’s relative rank. Considering multiple variants of the secretary problem, that vary from one another in their formulation and method of solution, we find that descriptive heuristics perform well only when the optimal solution prescribes a single threshold value. We show that a computational heuristic originally proposed as an approximate (...) solution to a single variant of the secretary problem performs equally well in many other variants where the optimal solution prescribes multiple threshold values that gradually relax the criterion for stopping the search. Finally, we propose a new heuristic with near optimal performance in a competitive or strategic variant of the secretary problem with multiple employers competing with one another to hire job applicants. Both heuristics share a simple computational component: the ratio of the number of interviewed applicants to the number of those remaining to be searched. We present the subgame-perfect Nash equilibrium for this competitive variant and an algorithm for its computation. (shrink)

We develop a logical system that captures two different interpretations of what extensive games model, and we apply this to a long-standing debate in game theory between those who defend the claim that common knowledge of rationality leads to backward induction or subgame perfect (Nash) equilibria and those who reject this claim. We show that a defense of the claim à la Aumann (1995) rests on a conception of extensive game playing as a one-shot event in combination (...) with a principle of rationality that is incompatible with it, while a rejection of the claim à la Reny (1988) assumes a temporally extended, many-moment interpretation of extensive games in combination with implausible belief revision policies. In addition, the logical system provides an original inductive and implicit axiomatization of rationality in extensive games based on relations of dominance rather than the usual direct axiomatization of rationality as maximization of expected utility. (shrink)

Miyagawa (Games and Economics Behavior 41(2), 292–308 [2002]) provides a simple extensive game form that implements a large class of two-agent bargaining solutions in subgame-perfect equilibrium. This class includes all of the Nash, Kalai–Smorodinsky, and relative utilitarian solutions. This note extends Miyagawa’s result to multi-agent bargaining problems.

A repulsive core force is derived which, assuming π mesons are the field particles, gives binding energies in good agreement with binding energies per nucleon of heavy nuclei. The physical model consists of a field of relatively short range, in which emission of a π meson by a nucleon and subsequent absorption by a neighboring nucleon is equivalent to a potential well. The binding energy at the equilibrium spacing of the nucleons is the self-energy of the π mesons, which (...) is calculated by analogy with classical electric fields. (shrink)

The extent to which trust prevails can be measured by the subjective probability with which an agent expects another one to act in desired ways. An agent´s trust in other agents forms in repeated social interactions which typically have the structure of an elementary game of trust. The process of trust formation in such interactions may be described by a reputation function. It is argued that in view of real world processes of trust formation any adequate reputation function must satisfy (...) certain conditions. A simple model conforming to these conditions is presented. Analysing this example it is shown that there is a co-operative Nash equilibrium in a trust supergame which is in line with the basic conditions of trust formation. However, it is also proved that no process of trust formation can be reasonably similar to real world mechanisms and at the same time lead to subgame perfect equilibria in a trust supergame. (shrink)

This paper reconsiders Rubinstein's alternating-offer bargaining game with complete information. We define rationalizability and trembling- hand rationalizability (THR) for multi-stage games with observed actions. We show that rationalizability does not exclude perpetual disagreement or delay, but that THR implies a unique solution. Moreover, this unique solution is the unique subgame perfect equilibrium (SPE). Also, we reconsider an extension of Rubinstein's game where a smallest money unit is introduced: THR rules out the non-uniqueness of SPE in some particular (...) case. Finally, we investigate the assumption of boundedly rational players. Perpetual disagreement is excluded, but not delay. Furthermore, we cannot use the asymmetric Nash bargaining solution as an approximation of the alternating-offer bargaining model once the players are boundedly rational ones. (shrink)

Backward induction was one of the earliest methods developed for solving finite sequential games with perfect information. It proved to be especially useful in the context of Tom Schelling’s ideas of credible versus incredible threats. BI can be also extended to solve complex games that include an infinite number of actions or an infinite number of periods. However, some more complex empirical or experimental predictions remain dramatically at odds with theoretical predictions obtained by BI. The primary example of such (...) a troublesome game is Centipede. The problems appear in other long games with sufficiently complex structure. BI also shares the problems of subgame perfect equilibrium and fails to eliminate certain unreasonable Nash equilibria. (shrink)

In political economy, the seminal contribution of the Baron–Ferejohn bargaining model constitutes an important milestone for the study of legislative policy making. In this paper, we analyze a particular equilibrium characteristic of this model, equilibrium uniqueness. The Baron–Ferejohn model yields a class of payoff-unique stationary subgame perfect equilibria in which players’ equilibrium strategies are not uniquely determined. We first provide a formal proof of the multiplicity of equilibrium strategies. This also enables us to establish (...) some important properties of SSPE. We then introduce veto players into the original Baron–Ferejohn model. We state the conditions under which the new model has a unique SSPE not only in terms of payoffs but also in terms of players’ equilibrium strategies. (shrink)

For the past 20 years or so the literature on noncooperative games has been centered on the search for an equilibrium concept that expresses the notion of rational behavior in interactive situations. A basic tenet in this literature is that if a “rational solution” exists, it must be a Nash equilibrium. The consensus view, however, is that not all Nash equilibria can be accepted as rational solutions. Consider, for example, the game of Figure 1.

In game theory, the word ‘game’ is used to describe any interdependence between interacting parties, and the Nash equilibrium is a prominent tool for analysing such interactions. I argue that the concept of the Nash equilibrium may also be used in non-gaming contexts. An individual is in a Nash equilibrium if his or her beliefs are consistent with his or her actions. Given that discordance between beliefs and behaviour is a typical cause of psychiatric (...) disorders, individuals who are not in a Nash equilibrium are likely to be affected by such disorders. In this regard, the concept of the Nash equilibrium could aptly be introduced into the medical practitioner’s vocabulary for describing a patient’s mental health status. (shrink)

The prelims comprise: Strategic Interaction Nash Equilibrium Normal‐form Refinements Games in Extensive Form Extensive‐form Refinements Selection by Evolution Notes.

How does Bayesian reasoning support participation in a game's Nash equilibrium? This paper provides an answer.

Inspired by Shalev’s model of loss aversion, we investigate the effect of loss aversion on a bimatrix game where the payoffs in the bimatrix game are characterized by triangular fuzzy variables. First, we define three solution concepts of credibilistic loss aversion Nash equilibria, and their existence theorems are presented. Then, three sufficient and necessary conditions are given to find the credibilistic loss aversion Nash equilibria. Moreover, the relationship among the three credibilistic loss aversion Nash equilibria is discussed (...) in detail. Finally, for2×2bimatix game with triangular fuzzy payoffs, we investigate the effect of loss aversion coefficients and confidence levels on the three credibilistic loss aversion Nash equilibria. It is found that an increase of loss aversion levels of a player leads to a decrease of his/her own payoff. We also find that the equilibrium utilities of players are decreasing as their own confidence levels when players employ the optimistic value criterion. (shrink)

Strong Nash equilibrium and coalition-proof Nash equilibrium rely on the idea that players are allowed to form coalitions and make joint deviations. Both of these notions consider cases in which any coalition can be formed. Accordingly, there may arise “conflicts of interest” that prevent a player from choosing an action that simultaneously meets the requirements of two coalitions to which he or she belongs. Here, we address this observation by studying an organizational framework such that the (...) coalitional structure is motivated by real-life examples where players cannot form some coalitions and formulated in such a way that no conflicts of interest remain. We define an organization as an ordered collection of partitions of the player set such that any partition is coarser than the partitions that precede it. For any given organization, we introduce the notion of organizational Nash equilibrium. We analyze the existence of equilibrium in a subclass of games with strategic complementarities and illustrate how the proposed notion refines the set of Nash equilibria in some examples of normal form games. (shrink)

The paper argues that the Nash Equilibrium Refinement Programme was less successful than its competitor, the Epistemic Programme. The prime criterion of success is the extent to which the programmes were able to reach the key objective guiding non-cooperative game theory for much of the twentieth century, namely, to develop a complete characterisation of the strategic rationality of economic agents in the form of the ultimate solution concept for any normal form and extensive game. The paper explains this (...) in terms of unjustified degrees of mathematisation in Nash Equilibrium Refinement Programme. While this programme’s mathematical models were often inspired by purely mathematical concerns rather than the economic phenomena they were intended to model, the Epistemic Programme’s models were developed with a keen eye to the role beliefs and desires play in strategic interaction between rational economic agents playing games; that is, their interactive epistemology. The Epistemic Programme succeeded in developing mathematical models formalising aspects of strategic interaction that remained implicit in the Nash Equilibrium Refinement Programme owing to an unjustified degree of mathematisation. As a result, the Epistemic Programme is the more successful theory.Keywords: Epistemic Programme; Game theory; Interactive epistemology; Mathematisation; Nash equilibrium. (shrink)

Game theory is often used to explain behavior. Such explanations often proceed by demonstrating that the behavior in question is a Nash equilibrium. Agents are in Nash equilibrium if each agent’s strategy maximizes her payoff given her opponents’ strategies. Nash equilibriums are fundamentally static, but it is usually assumed that equilibriums will be the outcome of a dynamic process of learning or evolution. This article demonstrates that, even in the most simple setting, this need not (...) be true. In two-strategy games with just a single equilibrium, a family of imitative learning dynamics does not lead to equilibrium. (shrink)

Game theory is usually considered applied mathematics, but a few game‐theoretic results, such as Borel determinacy, were developed by mathematicians for mathematics in a broad sense. These results usually state determinacy, i.e., the existence of a winning strategy in games that involve two players and two outcomes saying who wins. In a multi‐outcome setting, the notion of winning strategy is irrelevant yet usually replaced faithfully with the notion of (pure) Nash equilibrium. This article shows that every determinacy result (...) over an arbitrary game structure, e.g., a tree, is transferable into existence of multi‐outcome (pure) Nash equilibrium over the same game structure. The equilibrium‐transfer theorem requires cardinal or order‐theoretic conditions on the strategy sets and the preferences, respectively, whereas counter‐examples show that every requirement is relevant, albeit possibly improvable. When the outcomes are finitely many, the proof provides an algorithm computing a Nash equilibrium without significant complexity loss compared to the two‐outcome case. As examples of application, this article generalises Borel determinacy, positional determinacy of parity games, and finite‐memory determinacy of Muller games. (shrink)

The noema is one of the most daring and controversial concept of the Husserlian theory of intentionality. It was first introduced by Husserl in 1912, within some research manuscripts, but was only fully developed in Ideen. In this paper I claim that the noema is an ambiguous notion, the result of a theoretical operation, the epoché, whose aim is contradictory. In an effort to keep open the epoché, and therefore maintain distance with respect to every transcendent object, Husserl is forced (...) to multiply intentional objects and complicate the notions of sense and noema. Given that, I propose to overcome the paradoxes of noema through the language of game theory. Game theory offers a very fruitful descriptive model that allows us to save the original Husserlian approach without the contradictions of the epoché. For this reason, I propose to re-interpret intentionality as social game, and the noema as Nash equilibrium. By replying to possible objections, I will show that this approach gives us many theoretical advantages. The general aim of the paper is a global reformulation of the phenomenalogical method. (shrink)

The most important analytical tool in non-cooperative game theory is the concept of a Nash equilibrium, which is a collection of possibly mixed strategies, one for each player, with the property that each player's strategy is a best reply to the strategies of the other players. If we do not go into normative game theory, which concerns itself with the recommendation of strategies, and focus instead entirely on the positive theory of prediction, two alternative interpretations of the (...) class='Hi'>Nash equilibrium concept are predominantly available.In the more traditional one, a Nash equilibrium is a prediction of actual play. A game may not have a Nash equilibrium in pure strategies, and a mixed strategy equilibrium may be difficult to incorporate into this interpretation if it involves the idea of actual randomization over equally good pure strategies. In another interpretation originating from Harsanyi, see also Rubinstein, and Aumann and Brandenburger, a Nash equilibrium is a ‘consistent’ collection of probabilistic expectations, conjectures, on the players. It is consistent in the sense that for each player each pure strategy, which has positive probability according to the conjecture about that player, is indeed a best reply to the conjectures about others. (shrink)

This paper suggests that Nash equilibrium can be interpreted as a conceptual tool for addressing comparative-statics issues on an aggregate level. Together with additional assumptions, Nash equilibrium helps generate testable predictions about changes or differences of behavior induced by changes or differences of exogenous variables. However, Nash-equilibrium behavior, as such, is not subject to empirical testing. Instead, widespread and persistent behavior is modeled as a Nash equilibrium by assumption. The paper argues that (...) this interpretation is in line with the traditional use of an equilibrium concept in price theory. The suggested interpretation is illustrated with the model of public-goods provision. The paper compares this interpretation of Nash equilibrium with other interpretations and discusses its limits. (shrink)

This paper investigates the existence of pure strategy Nash equilibria in discontinuous and nonquasiconcave games. We introduce a new notion of continuity, called weakly continuous security, which is weaker than the most known weak notions of continuity, including the surrogate point secure of SSYM game of Carbonell-Nicolau and Mclean (Econ Theory, 2018a), the continuous security of Barelli and Meneghel (Econometrica 81:813–824, 2013), C-security of McLennan et al. (Econometrica 79:1643–1664 2011), generalized weakly transfer continuity of Nessah (Economics 47:659–662, 2011), generalized (...) better-reply security of Carmona (Econ Theory 48:1–4, 2011), Barelli and Soza (On the existence of Nash equilibria in discontinuous and qualitative games, 2009), Barelli and Meneghel (Econometrica 81:813–824, 2013), lower single deviation property of Reny (Further results on the existence of Nash equilibria in discontinuous games. University of Chicago, 2009), better-reply security of Reny (Econometrica 67:1029–1056, 1999) and the results of Prokopovych (Econ Theory 48:5–16, 2011, Econ Theory 53:383–402, 2013) and Carmona (J Econ Theory 144:1333–1340, 2009). We show that a compact, convex and weakly continuous secure Hausdorff locally convex topological vector space game has a pure strategy Nash equilibrium. Moreover, it holds in a large class of discontinuous games. (shrink)

Metalogic is an open-ended cognitive, formal methodology pertaining to semantics and information processing. The language that mathematizes metalogic is known as metalanguage and deals with metafunctions purely by extension on patterns. A metalogical process involves an effective enrichment in knowledge as logical statements, and, since human cognition is an inherently logic–based representation of knowledge, a metalogical process will always be aimed at developing the scope of cognition by exploring possible cognitive implications reflected on successive levels of abstraction. Indeed, it is (...) basically impracticable to maintain logic–and–metalogic without paying heed to cognitive theorizing. For it is cognitively irrelevant whether possible conclusions are deduced from some premises before the premises are determined to be true or whether the premises themselves are determined to be true first and, then, the conclusions are deduced from them. In this paper we consider the term metalogic as inherently embodied under the framework referred to as cognitive science and mathematics. We propose a metalogical interpretation of Arrow’s impossibility theorem and, to that end, choice theory is understood as a mental course of action dealing with logic and metalogic issues, in which a possible mathematical approach to model a mental course of action is adopted as a systematic operating method. Nevertheless, if we look closely at the core of Arrow’s impossibility theorem in terms of metalogic, a second fundamental contribution to this framework is represented by the Nash equilibrium. As a result of the foregoing, therefore, we prove the metalogical equivalence between Arrow’s impossibility theorem and the existence of the Nash equilibrium. More specifically, Arrow’s requirements correspond to the Nash equilibrium for finite mixed strategies with no symmetry conditions. To demonstrate this proof, we first verify that Arrow’s set and Nash’s set are isomorphic to each other, both sets being under stated conditions of non–symmetry. Then, the proof is completed by virtue of category theory. Indeed, the two sets are categories that correspond biuniquely to one another and, thus, it is possible to define a covariant functor that preserves their mutual structures. According to this, we show the proof–dedicated metalanguage as a precursor to a special equivalence theorem. (shrink)

In classical game theory the idea that players randomize between their actions according to a particular optimal probability distribution has always been viewed as puzzling. In this paper, we establish a fundamental connection between n-person normal form games and quantum mechanics, which eliminates the conceptual problems of these random strategies. While the two theories have been regarded as distinct, our main theorem proves that if we do not give any other piece of information to a player in a game, than (...) the payoff matrix—the axiom of “no-supplementary data” holds—then the state of mind of a rational player is algebraically isomorphic to a pure quantum state. The “no supplementary data” axiom is captured in a Lukasiewicz’s three-valued Kripke semantics wherein statements about whether a strategy or a belief of a player is rational are initially indeterminate i.e. neither true, nor false. As a corollary, we show that in a mixed Nash equilibrium, the knowledge structure of a player implies that probabilities must verify the standard “Born rule” postulate of QM. The puzzling “indifference condition” wherein each player must be rationally indifferent between all the pure actions of the support of his equilibrium strategy is resolved by his state of mind being described by a “quantum superposition” prior a player is asked to make a definite choice in a “measurement”. Finally, these results demonstrate that there is an intrinsic limitation to the predictions of game theory, on a par with the “irreducible randomness” of quantum physics. (shrink)

In the present paper, we consider a bio-economic equilibrium model which describes the dynamics of a fish population fished by several fishermen seeking to maximize their profits. Each fisherman tries to find the fishing effort which maximizes his profit at biological equilibrium without any consultation with others, but all of them have to respect two constraints: (1) the sustainable management of the resources ; and (2) the preservation of the biodiversity. With all these considerations, our problem leads to (...) a generalized Nash equilibrium problem. The objective is to show that even when a fisherman i\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$i$$\end{document} provides a fishing effort equal to twice the fishing effort of a fisherman j\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$j$$\end{document}, then the profit of fisherman i\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$i$$\end{document} is not necessarily double that of fisherman j\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$j$$\end{document}. (shrink)

Donald Capps (2009: 145) suggested the hypothesis that “the Nash equilibrium is descriptive of the normal brain, whereas the game theory formulated by John van Neumann, which Nash’s theory challenges, is descriptive of the schizophrenic brain”. The paper offers arguments in its favor. They are from psychiatry, game theory, set theory, philosophy and theology. The Nash equilibrium corresponds to wholeness, stable emergent properties as well as to representing actual infinity on a material, limited and finite (...) organ as a human brain. (shrink)

We provide eductive foundations for the concept of forward induction, in the class of games with an outside option. The formulation presented tries to capture in a static notion the rest point of an introspective process, achievable from some restricted preliminary beliefs. The former requisite is met by requiring the rest point to be a Nash equilibrium that yields a higher payoff than the outside option. With respect to the beliefs, we propose the Incentive Dominance Criterion. Players should (...) consider one action more likely than another whenever the former is better than getting the outside option for more conjectures over his rival's actions. We apply this model to the case where the subgame is a coordination game with a conflict between payoff dominance and risk dominance. Our results provide support for dominance solvability, but not for Van Damme's notion of forward induction. We show how the forward induction logic helps to select the Pareto dominant equilibrium. This is the case whenever player 1's act of giving up the outside option reverses the incentive dominance relations among 1's pure actions in the subgame. (shrink)

Gupta has proposed a definition of strategic rationality cast in the framework of his revision theory of truth. His analysis, relative to a class of normal form games in which all players have a strict best reply to all other players’ strategy profiles, shows that game-theoretic concepts have revision-theoretic counterparts. We extend Gupta’s approach to deal with normal form games in which players’ may have weak best replies. We do so by adapting intuitions relative to Nash equilibrium refinements (...) to the revision-theoretic framework. We prove that there is a precise equivalence between trembling-hand perfect equilibria in two-player normal games and a revision-theoretic property. We then introduce lexicographic choice of action as a way to represent players’ expectations, which allows our analysis to reach full generality. Finally, we provide an example of the versatility of revision theory as applied to strategic interaction by formalizing a risk-and-compensation procedure of strategic choice in the revision-theoretic framework. (shrink)

We start by considering the Alternate Strike (AS) scheme, a real-life arbitration scheme where two parties select an arbitrator by alternately crossing off at each round one name from a given panel of arbitrators. We find out that the AS scheme is not invariant to “bad” alternatives. We then consider another alternating-move scheme, the Voting by Alternating Offers and Vetoes (VAOV) scheme, which is invariant to bad alternatives. We fully characterize the subgame perfect equilibrium outcome sets of (...) these above two schemes in terms of the rankings of the parties over the alternatives only. We also identify some of the typical equilibria of these above two schemes. We then analyze two additional alternating-move schemes in which players’ current proposals have to either honor or enhance their previous proposals. We show that the first scheme’s equilibrium outcome set coincides with that of the AS scheme, and the equilibrium outcome set of the second scheme coincides with that of the VAOV scheme. Finally, it turns out that all schemes’ equilibrium outcome sets converge to the Equal Area solution’s outcome of cooperative bargaining problem, if the alternatives are distributed uniformly over the comprehensive utility possibility set and as the number of alternatives tends to infinity. Journal of Economic Literature Classification Number: C72. (shrink)

We propose a new game to strategically justify the constrained equal awards rule in claims problems. Our game is “procedurally fair” and “multilateral”. In addition, even if claimants cannot reach an agreement in any period, they can renegotiate in the next period. We show that, for each claims problem, the awards vector chosen by the constrained equal awards rule achieved at period 1 is the unique subgame perfect equilibrium outcome of the game.

This paper analyzes a model of sequential parimutuel betting described as a two-horse race with a finite number of noise bettors and a finite number of strategic and symmetrically informed bettors. For generic objective probabilities that the favorite wins the race, a unique subgame perfect equilibrium is characterized. Additionally, two explanations for the favorite–longshot bias—according to which favorites win more often than the market's estimate of their winning chances imply—are offered. It is shown that this robust anomalous (...) empirical regularity might be due to the presence of transaction costs and/or to strategic bettors' subjective attitude to probabilities. (shrink)

In contrast to Axelrod’s advice “don’t be envious” it is argued that the emotion of envy may enhance cooperation. TIT FOR TAT does exhibit a certain degree of envy. But, it does so in inconsistent ways. Two variants of TIT FOR TAT are introduced and their strategic properties are analyzed. Both generate the very same actual play as TIT FOR TAT in a computer tournament without noise. However, if noise is introduced they display some greater degree of stability. This is (...) due to the fact that they form, in a prisoner’s dilemma supergame with suitable parameters, an equilibrium with themselves that is subgame perfect or (in case of the first strategy) close to subgame perfect. It is additionally argued that these strategies are exceptionally clear and comprehensible to others in that they conform to well known real live behavior patterns. (shrink)

Two players bargain to select a utility allocation in some set X⊂R+2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$X\subset {\mathbb {R}}_+^2$$\end{document}. Bargaining takes place in infinite discrete time, where each period t is divided into two sub-periods. In the first sub-period, the players play a simultaneous-move game to determine that period’s proposer, and bargaining takes place in the second sub-period. Rejection triggers a one-period delay and move to t+1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$t+1$$\end{document}. For (...) every x∈X∩R++2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$x\in X\cap {\mathbb {R}}^2_{++}$$\end{document}, there exists a cutoff δ<1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\delta <1$$\end{document}, such that if at least one player has a discount factor above δ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\delta $$\end{document}, then for every y∈X\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$y\in X$$\end{document} that satisfies y≥x\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$y\ge x$$\end{document} there exists a subgame perfect equilibrium with immediate agreement on y. The equilibrium is supported by “dictatorial threats.” These threats can be dispensed with if X is the unit simplex and the target-vector is Pareto efficient. The results can be modified in a way that allows for arbitrarily long delays in equilibrium. (shrink)

A dynamic choice mechanism (e.g. sophisticated choice) is a rule transforming a sequence of orderings into actual choices. The dynamic choice literature analysing such rules is restricted to the case of strict preferences. The present paper allows also for weak preferences. This generalisation is necessary since in dynamic utility models decision makers are typically endowed with orderings which can be represented by continuous utility functions. Such a representation, however, requires the orderings to be a weak ordering. In the presence of (...) weak orderings the classical choice mechanisms have to be re-examined. This paper develops and analyses new choice mechanisms and relates them to the notion of a subgame-perfect equilibrium. (shrink)

We consider a spatial bargaining model where players collectively choose a facility location on a two-dimensional rectilinear distance space through bargaining using the unanimity rule. We show that as players become infinitely patient, their stationary subgame perfect equilibrium utilities converge to the utilities that satisfy the lexicographic maximin utility criterion introduced by Sen.

We show that for solvable games, the calculation of the strategies which survive iterative elimination of dominated strategies in normal games is equivalent to the calculation of the backward induction outcome of some extensive game. However, whereas the normal game form does not provide information on how to carry out the elimination, the corresponding extensive game does. As a by-product, we conclude that implementation using a subgame perfect equilibrium of an extensive game with perfect information is (...) equivalent to implementation through a solution concept which we call guided iteratively elimination of dominated strategies which requires a uniform order of elimination. Journal of Economic Literature Classification Number: C72. (shrink)

The order of stages in a multistage game is often interpreted by looking at earlier stages as involving more long term decisions. For the purpose of making this interpretation precise, the notion of a delay supergame of a bounded multistage game is introduced. A multistage game is bounded if the length of play has an upper bound. A delay supergame is played over many periods. Decisions on all stages are made simultaneously, but with different delays until they become effective. The (...) earlier the stage the longer the delay. A subgame perfect equilibrium of a bounded multistage game generates a subgame perfect equilibrium in every one of its delay supergames. This is the first main conclusion of the paper. A subgame perfect equilibrium set is a set of subgame perfect equilibria all of which yield the same payoffs, not only in the game as a whole, but also in each of its subgames. The second xmain conclusion concerns multistage games with a unique subgame perfect equilibrium set and their delay supergames which are bounded in the sense that the number of periods is finite. If a bounded multistage game has a unique subgame perfect equilibrium set, then the same is true for every one of its bounded delay supergames. Finally the descriptive relevance of multistage game models and their subgame perfect equilibria is discussed in the light of the results obtained. (shrink)

We consider an implementation of the discounted Shapley value. We modify the Hart–Mas-Colell model such that each player discounts future payoffs and proposes not only an allocation, but also a coalition. We show that the discounted Shapley value is supported by each stationary subgame perfect equilibrium of the modified game such that in each subgame, the coalition that consists of all active players immediately forms. We also provide conditions for such a stationary subgame perfect (...) equilibrium to exist. (shrink)

We show that for solvable games, the calculation of the strategies which survive iterative elimination of dominated strategies in normal games is equivalent to the calculation of the backward induction outcome of some extensive game. However, whereas the normal game form does not provide information on how to carry out the elimination, the corresponding extensive game does. As a by-product, we conclude that implementation using a subgame perfect equilibrium of an extensive game with perfect information is (...) equivalent to implementation through a solution concept which we call guided iteratively elimination of dominated strategies which requires a uniform order of elimination. Journal of Economic Literature Classifica-. (shrink)